Yes! The above is totally correct that is because if we define

and

by

then the continuity of

easily implies the continuity of

. Thus, one notes that since

is Hausdorff that

is closed and

and so

is closed. So, if

is dense then

and so

. But! By definition this means that

or that

!

Maybe more intuitive is to notice that

is a topological group (with the usual addition and usual topology). And since given a topological group

the map

given by

g,h)\mapsto gh^{-1}" alt="\alpha

g,h)\mapsto gh^{-1}" /> is continuous. Thus, one notes that

is continuous and thus so is

with

. So, noticing that

and thus the above claim is true if

is closed in

. And since

is closed in

the conclusion follows!

(notice that the above paragraph is a farce since

)